The principality club reclaimed top spot with 18 points from eight games, ahead of Paris St Germain on goal difference after the French champions beat Toulouse 2-0 on Saturday.
Olympique Marseille, who visit Borussia Dortmund in the Champions League on Tuesday, are third on 17.
Antoine Devaux had put the hosts ahead after just four minutes with a neat header.
Moutinho levelled for the big-spenders only 11 minutes later, becoming the third player to score for Monaco whose previous 13 goals were credited to Radamel Falcao (7) and Emmanuel Riviere (6).
The Portuguese, who had already set up three goals in four appearances, took over a 25-metre free kick and smartly bent the ball in at the near post after seeing that keeper Kossi Agassa, expecting a cross, had deserted the goal.
Monaco had their chances in an entertaining match but it was Reims who went the closest to scoring again when Floyd Ayite fired the ball on to the crossbar around the hour mark.